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Which of the following equations has two distinct real roots?

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2 x 2 − 32 x + 94 = 0

x 2 + x − 5 = 0

x 2 + 3 x + 22 = 0

5 x 2 − 3 x + 1 = 0

answer is B.

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Detailed Solution

Given equations are

2 x 2 − 32 x + 94 = 0,

x 2 + x − 5 = 0,

x 2 + 3 x + 22 = 0,

and

5 x 2 − 3 x + 1 = 0.

For a quadratic equation

a x 2 + b x + c = 0

a ≠ 0

, there exist two distinct real roots if the discriminant,

D = b 2 − 4 a c > 0

.
On comparing

2 x 2 − 32 x + 94 = 0

with

a x 2 + b x + c = 0

, we get

a = 2, b = − 32, c = 94

.
So, the discriminant will be,

D = b 2 − 4 a c = (32) 2 − 4 (2) 94 = 18 − 18 = 0

The equation

2 x 2 − 32 x + 94 = 0

has equal and real roots.
On comparing

x 2 + x − 5 = 0

with

a x 2 + b x + c = 0

, we get,

a = 1, b = 1, c = - 5

.
So, the discriminant will be,

D = b 2 − 4 a c = (1) 2 − 4 (1) − 5 = 1 + 20 = 21 > 0

The equation

x 2 + x − 5 = 0

has two distinct real roots.
On comparing

x 2 + 3 x + 22 = 0

with

a x 2 + b x + c = 0

, we get

a = 1, b = 3, c = 22

.
So, the discriminant will be,

D = b 2 − 4 a c = (3) 2 − 4 (1) 22 = 9 − 82 < 0

The equation

x 2 + 3 x + 22 = 0

has no real roots.
On comparing

5 x 2 − 3 x + 1 = 0

with

a x 2 + b x + c = 0

, we get,

a = 5, b = - 3, c = 1

.
So, the discriminant will be,

D = b 2 − 4 a c = (3) 2 − 4 (1) 5 = 9 − 20 = − 11 < 0

The equation

5 x 2 − 3 x + 1 = 0

has no real roots.
Thus, the only equation with real and distinct roots is

x 2 + x − 5 = 0

.
Hence, the correct option is 2.

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